Transcript 5-5 Similar Figures - Maple Heights City Schools

5-5
5-5 Similar
SimilarFigures
Figures
Warm Up
Problem of the Day
Lesson Presentation
Course
Course
33
5-5 Similar Figures
Warm Up
Solve each proportion.
1. 3 = b
9
30
3. p = 4
9
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12
b = 10
p=3
2.
y = 56
35
5
4. 28 = 56
26
m
y=8
m = 52
5-5 Similar Figures
Problem of the Day
A rectangle that is 10 in. wide and 8 in. long
is the same shape as one that is 8 in. wide
and x in. long. What is the length of the
smaller rectangle?
6.4 in.
Course 3
5-5 Similar Figures
Learn to determine whether figures are
similar, to use scale factors, and to find
missing dimensions in similar figures.
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5-5 Similar Figures
Vocabulary
similar
congruent angles
scale factor
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5-5 Similar Figures
Similar figures have the same shape, but not
necessarily the same size. Two triangles are similar if
the ratios of the lengths of corresponding sides are
equivalent and the corresponding angles have equal
measures. Angles that have equal measures are
called congruent angles.
43°
43°
43°
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5-5 Similar Figures
Additional Example 1: Identifying Similar Figures
Which rectangles are similar?
Since the three figures are all rectangles, all the
angles are right angles. So the corresponding
angles are congruent.
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5-5 Similar Figures
Additional Example 1 Continued
Compare the ratios of corresponding sides to see if they are equal.
length of rectangle J
length of rectangle K
10 ? 4
5 =2
20 = 20
width of rectangle J
width of rectangle K
The ratios are equal. Rectangle J is similar to
rectangle K. The notation J ~ K shows similarity.
length of rectangle J
length of rectangle L
10 ? 4
12 = 5
width of rectangle J
width of rectangle L
50  48
The ratios are not equal. Rectangle J is not similar to
rectangle L. Therefore, rectangle K is not similar to
rectangle L.
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5-5 Similar Figures
Check It Out: Example 1
Which rectangles are similar?
8 ft
A
4 ft
6 ft
B
3 ft
5 ft
C
2 ft
Since the three figures are all rectangles, all the
angles are right angles. So the corresponding
angles are congruent.
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5-5 Similar Figures
Check It Out: Example 1 Continued
Compare the ratios of corresponding sides to see if they are equal.
length of rectangle A
length of rectangle B
8 ? 4
6 =3
24 = 24
width of rectangle A
width of rectangle B
The ratios are equal. Rectangle A is similar to
rectangle B. The notation A ~ B shows similarity.
length of rectangle A
length of rectangle C
8 ? 4
5= 2
width of rectangle A
width of rectangle C
16  20
The ratios are not equal. Rectangle A is not similar
to rectangle C. Therefore, rectangle B is not similar
to rectangle C.
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5-5 Similar Figures
The ratio formed by the corresponding
sides is the scale factor.
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5-5 Similar Figures
Additional Example 2A: Using Scale Factors to Find
Missing Dimensions
A picture 10 in. tall and 14 in. wide is to be
scaled to 1.5 in. tall to be displayed on a Web
page. How wide should the picture be on the
Web page for the two pictures to be similar?
1.5 = 0.15
10
Divide the height of the scaled
picture by the corresponding height
of the original picture.
14 • 0.15
Multiply the width of the original
picture by the scale factor.
2.1
Simplify.
The picture should be 2.1 in. wide.
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5-5 Similar Figures
Additional Example 2B: Using Scale Factors to Find
Missing Dimensions
A toy replica of a tree is 9 inches tall. What is
the length of the tree branch, if a 6 ft tall tree
has a branch that is 60 in. long?
Divide the height of the replica tree
9 = .125 by the height of the 6 ft tree in
72
inches, to find the scale factor.
60 • .125 Multiply the length of the original tree
branch by the scale factor.
7.5
Simplify.
The toy replica tree branch should be 7.5 in long.
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5-5 Similar Figures
Check It Out: Example 2A
A painting 40 in. tall and 56 in. wide is to be
scaled to 10 in. tall to be displayed on a
poster. How wide should the painting be on
the poster for the two pictures to be similar?
10 = 0.25
40
Divide the height of the scaled picture
by the corresponding height of the
original picture.
56 • 0.25
Multiply the width of the original
picture by the scale factor.
14
Simplify.
The picture should be 14 in. wide.
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5-5 Similar Figures
Check It Out: Example 2B
A toy replica of a tree is 6 inches tall. What is
the length of the tree branch, if a 8 ft tall tree
has a branch that is 72 in. long?
6 = .0625
96
Divide the height of the replica tree
by the height of the 6 ft tree in
inches, to find the scale factor.
72 • .0625
Multiply the length of the original tree
branch by the scale factor.
4.5
Simplify.
The toy replica tree branch should be 4.5 in long.
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5-5 Similar Figures
Additional Example 3: Using Equivalent Ratios to
Find Missing Dimensions
A T-shirt design includes an isosceles triangle
with side lengths 4.5 in, 4.5 in., and 6 in. An
advertisement shows an enlarged version of the
triangle with two sides that are each 3 ft. long.
What is the length of the third side of the
triangle in the advertisement?
4.5 in. = 6 in.
3 ft
x ft
Set up a proportion.
4.5 in. • x ft = 3 ft • 6 in.
Find the cross products.
4.5 in. • x ft = 3 ft • 6 in.
Divide out the units.
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5-5 Similar Figures
Helpful Hint
Draw a diagram to help you visualize the
problems
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5-5 Similar Figures
Additional Example 3 Continued
4.5x = 3 • 6
Cancel the units.
4.5x = 18
Multiply.
x = 18 = 4
4.5
Solve for x.
The third side of the triangle is 4 ft long.
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5-5 Similar Figures
Check It Out: Example 3
A flag in the shape of an isosceles triangle with
side lengths 18 ft, 18 ft, and 24 ft is hanging on
a pole outside a campground. A camp t-shirt
shows a smaller version of the triangle with two
sides that are each 4 in. long. What is the length
of the third side of the triangle on the t-shirt?
18 ft = 24 ft
4 in.
x in.
Set up a proportion.
18 ft • x in. = 24 ft • 4 in. Find the cross products.
18 ft • x in. = 24 ft • 4 in. Divide out the units.
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5-5 Similar Figures
Check It Out: Example 3 Continued
18x = 24 • 4
Cancel the units.
18x = 96
Multiply.
x = 96  5.3
18
Solve for x.
The third side of the triangle is about 5.3 in.
long.
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5-5 Similar Figures
Lesson Quiz
Use the properties of similar figures to
answer each question.
1. A rectangular house is 32 ft wide and 68 ft long.
On a blueprint, the width is 8 in. Find the length
on the blueprint. 17 in.
2. Karen enlarged a 3 in. wide by 5 in. tall photo into
a poster. If the poster is 2.25 ft wide, how tall is
it? 3.75 ft
3. Which rectangles are
similar?
A and B are similar.
Course 3